The Derivative of the Tangent Function | Understanding d tan(x) and its Relationship with the Secant Function

d tan x

The expression “d tan x” represents the derivative of the tangent function with respect to the variable x

The expression “d tan x” represents the derivative of the tangent function with respect to the variable x. The derivative of the tangent function can be written as:

d tan(x) = sec^2(x) dx

Here, sec^2(x) represents the square of the secant function, which is defined as the reciprocal of the cosine function:

sec(x) = 1/cos(x)

Therefore, these two expressions can be used interchangeably:

d tan(x) = (1/cos^2(x)) dx

To find the derivative of tan(x), we can use the quotient rule. The quotient rule states that for functions f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

In the case of tan(x), g(x) = sin(x) and h(x) = cos(x). Taking the derivatives of sin(x) and cos(x), we have:

g'(x) = cos(x) and h'(x) = -sin(x)

Substituting these values into the quotient rule, we get:

d tan(x) = [(cos(x) * cos(x)) – (sin(x) * (-sin(x)))] / (cos(x))^2

Simplifying further:

d tan(x) = (cos^2(x) + sin^2(x)) / (cos^2(x))

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can simplify the expression:

d tan(x) = 1 / cos^2(x)

Therefore, the derivative of the tangent function with respect to x is equal to 1 divided by the square of the cosine of x.

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